Quantum circuits with reduced t gate count

ABSTRACT

Methods, systems and apparatus for producing quantum circuits with low T gate counts. In one aspect, a method for performing a temporary logical AND operation on two control qubits includes the actions of obtaining an ancilla qubit in an A-state; computing a logical-AND of the two control qubits and storing the computed logical-AND in the state of the ancilla qubit, comprising replacing the A-state of the ancilla qubit with the logical-AND of the two control qubits; maintaining the ancilla qubit storing the logical-AND of the two controls until a first condition is satisfied; and erasing the ancilla qubit when the first condition is satisfied.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation application of and claims priority toU.S. application Ser. No. 17/353,411, filed on Jun. 21, 2021, which is acontinuation of and claims priority to U.S. application Ser. No.16/644,657, filed on Mar. 5, 2020, which is a U.S. National StageApplication under 35 U.S.C. § 371 and claims the benefit ofInternational Patent Application Serial No. PCT/US2017/067577, filedDec. 20, 2017, which claims priority to Provisional Application Ser. No.62/556,163, filed Sep. 8, 2017. The disclosures of the priorapplications are considered part of and are incorporated by reference inthe disclosure of this application.

BACKGROUND

A quantum circuit is a model for quantum computation in which acomputation is a sequence of quantum logic gates—reversibletransformations on an n-qubit register.

SUMMARY

The subject matter of the present specification relates to technologiesfor producing quantum circuits, such as quantum circuits with low T gatecounts.

In general, one innovative aspect of the subject matter described inthis specification can be implemented in a method for performing atemporary Toffoli quantum logic gate on two control qubits and a targetqubit, the method comprising: obtaining an ancilla qubit in an A-state;computing a logical-AND of the two control qubits and storing thecomputed logical-AND in the state of the ancilla qubit, comprisingreplacing the A-state of the ancilla qubit with the logical-AND of thetwo control qubits; applying a CNOT quantum logic gate between (i) theancilla qubit storing the logical-AND of the two control qubits, and(ii) the target qubit, the ancilla qubit acting as a control qubit forthe CNOT quantum logic gate; providing the ancilla qubit storing thelogical-AND of the two control qubits as a resource for one or moreadditional operations; uncomputing the logical-AND of the two controlqubits, comprising recovering the A-state of the ancilla qubit byreplacing the state of the ancilla qubit storing the computedlogical-AND of the two control qubits with an A-state; and providing theancilla qubit in the recovered A-state as a resource for one or moreadditional operations.

The foregoing and other implementations can each optionally include oneor more of the following features, alone or in combination. In someimplementations providing the ancilla qubit in the A-state as a resourcefor one or more additional operations comprises providing the ancillaqubit in the A-state to perform a T gate.

In some implementations the method is used to perform a first temporaryToffoli quantum logic gate on two control qubits and a target qubit,wherein providing the ancilla qubit in the recovered A-state as aresource for one or more additional operations comprises providing theancilla qubit in the recovered A-state to perform a second temporaryToffoli quantum logic gate on two control qubits and a target qubit.

In some implementations computing a logical-AND of the two controlqubits and uncomputing the logical-AND of the two control qubitscomprises performing six T gates.

In some implementations computing a logical-AND of the two controlqubits and storing the computed logical-AND in the state of the ancillaqubit comprises: applying a CNOT gate between the ancilla qubit in the|A> state and a first control qubit; applying the Hermitian conjugate ofa T gate to the ancilla qubit; applying a CNOT gate between the ancillaqubit and a second control qubit; applying a T gate to the ancillaqubit; applying a CNOT gate between the ancilla qubit and the firstcontrol qubit; applying the Hermitian conjugate of a T gate to theancilla qubit; and applying a Hadamard gate to the ancilla qubit tostore the logical AND of the two control qubits in the state of theancilla qubit.

In some implementations the method further comprises applying a S gateto the ancilla qubit storing the logical-AND of the two control qubits.

In some implementations recovering the A-state of the ancilla qubit byreplacing the state of the ancilla qubit storing the computedlogical-AND of the two control qubits with an A-state comprises:applying a Hadamard gate to the ancilla qubit storing the logical-AND ofthe two control qubits; applying a T gate to the ancilla qubit; applyinga CNOT gate between the ancilla qubit and a first control qubit;applying the Hermitian conjugate of a T gate to the ancilla qubit;applying a CNOT gate between the ancilla qubit and a second controlqubit; applying a T gate to the ancilla qubit; and applying a CNOT gatebetween the ancilla qubit and the first control qubit to leave theancilla qubit in an |A> state.

In some implementations the method further comprises applying a S gateto the ancilla qubit.

In general, another innovative aspect of the subject matter described inthis specification can be implemented in a method for performing atemporary logical AND operation on two control qubits, the methodcomprising: obtaining an ancilla qubit in an A-state; computing alogical-AND of the two control qubits and storing the computedlogical-AND in the state of the ancilla qubit, comprising replacing theA-state of the ancilla qubit with the logical-AND of the two controlqubits; maintaining the ancilla qubit storing the logical-AND of the twocontrols until a first condition is satisfied; and erasing the ancillaqubit when the first condition is satisfied.

The foregoing and other implementations can each optionally include oneor more of the following features, alone or in combination. In someimplementations maintaining the ancilla qubit storing the logical-AND ofthe two controls until a first condition is satisfied comprisesproviding the ancilla qubit storing the logical-AND of the two controlqubits as a resource for one or more additional operations.

In some implementations the one or more additional operations compriseoperations that would otherwise be conditioned on the two controlqubits.

In some implementations erasing the ancilla qubit when the firstcondition is satisfied comprises erasing the ancilla qubit when the oneor more additional operations have been performed.

In some implementations erasing the ancilla qubit comprisestransitioning the ancilla into a state that is independent of the stateof the two control qubits and does not cause the two control qubits todecohere.

In some implementations erasing the ancilla qubit comprises applying ameasure-and-correct process.

In some implementations the measure-and-correct process comprises:applying a Hadamard quantum logic gate to the ancilla qubit; measuringthe ancilla qubit to generate a measurement result; in response todetermining that the generated measurement result indicates that the twocontrol qubits are both ON, applying a CZ gate.

In some implementations the measure-and-correct process comprisesClifford operations.

In some implementations the method further comprises correcting phaseerrors by applying an uncontrolled S gate to the ancilla qubit.

In some implementations computing the logical-AND of the two controlqubits comprises applying three T gates, optionally approximately inparallel.

The subject matter described in this specification can be implemented inparticular ways so as to realize one or more of the followingadvantages.

The presently described disclosure represents a significant and widelyapplicable improvement to the state of the art in synthesizing quantumcircuits with low T gate counts.

For example, for quantum circuits produced using previously knownmethods, addition operations typically have a T-count of 8n+O(1) with nrepresenting the number of qubits that the circuit operates on. Forquantum circuits produced using the presently disclosed techniques, theT-count of addition operations is halved to 4n+O(1). In particular, thepresently disclosed techniques includes a construction called atemporary AND gate that uses four T gates to store the logical-AND oftwo qubits into an ancilla qubit and zero T gates to later erase theancilla qubit. Temporary AND gates may be a useful tool when optimizingT-count, and can be applied to integer arithmetic, modular arithmetic,rotation synthesis, the quantum Fourier transform, Shor's algorithm,Grover oracles, and many other circuits. In addition, because T gatesdominate the cost of quantum computation based on the surface code, andthe temporary AND gate is widely applicable, the disclosed constructionsrepresent a significant reduction in projected costs of quantumcomputation.

Furthermore, the presently disclosed techniques further include theconstruction of an n-bit controlled adder circuit with T-count of8n+O(1), and a temporary adder that can be computed for the same cost asthe normal adder but whose result can be kept until later un-computedwithout using T gates.

Details of one or more implementations of the subject matter of thisspecification are set forth in the accompanying drawings and thedescription below. Other features, aspects, and advantages of thesubject matter will become apparent from the description, the drawings,and the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an exemplary circuit representation of a Toffoli construction.

FIG. 2 depicts an exemplary system for implementing temporary Toffoligates and logical AND operations.

FIG. 3 is a flow diagram of an exemplary process for indirectlyperforming a Toffoli gate on two control qubits and a target qubit.

FIG. 4 is an illustration of an exemplary quantum circuit for indirectlyperforming a Toffoli gate on two control qubits and a target qubit.

FIG. 5 is an illustration of an exemplary quantum circuit for indirectlyperforming multiple Toffoli gates on two control qubits and a targetqubit.

FIG. 6 is a flow diagram of an exemplary process for performing atemporary logical AND operation on two control qubits.

FIG. 7 is an illustration of an exemplary quantum circuit for performinga temporary logical AND operation on two control qubits.

FIG. 8 is an illustration of an exemplary quantum circuit forun-computing a temporary logical AND operation on two control qubits.

FIG. 9 is an illustration of a per-bit building block of an improvedadder construction with a T count of 4.

FIG. 10 is an illustration a 5 bit adder with a T count of 16.

DETAILED DESCRIPTION

The surface code is a quantum error correcting code that may operate ona two dimensional (2D) nearest-neighbor array of qubits and achieve athreshold error rate of approximately 1%. This makes the surface code alikely component in the architecture of future error corrected quantumcomputers, since 2D arrays of qubits with nearest-neighbor connectionsmay be implemented using many qubit technologies and other known errorcorrecting codes have lower thresholds or require stronger connectivity.

One downside of the surface code is that it has no cheap mechanism toapply non-Clifford operations such as T gates that perform 45 degreerotations around the Z axis of the Bloch sphere. Instead, T gates areperformed by distilling and consuming

$\left. \left. {\left. {\left. {❘A} \right\rangle = {\frac{1}{\sqrt{2}}\left( {❘0} \right.}} \right\rangle + {e^{\frac{i\pi}{4}}{❘1}}} \right\rangle \right)$

states. Consuming an |A

state to perform a T gate is simple, but distilling |A

states has significant cost. Because T gates are so expensive for thesurface code, and the surface code is a likely component of futurequantum computers, it may be advantageous to reduce the number of Tgates used by quantum circuits to perform certain quantum computingoperations.

This specification describes methods and constructions that may beimplemented in a quantum circuit on a quantum device for improving thenumber of T gates needed to perform a first Toffoli gate that is laterun-computed by a second Toffoli gate. The first Toffoli gate isperformed indirectly by targeting a clean ancilla qubit and then usingthe ancilla qubit to toggle the intended target. The ancilla qubit isnot un-computed and re-computed if it will be provided as a resource foradditional operations. If a T gate is used to compute or un-compute an|A

state, an |A

state is passed in or recovered. The ancilla qubit is un-computed bymeasurement of the ancilla qubit and application of a classicallycontrolled operation (also described as a measure-and-correct processherein).

In this specification, initializing the ancilla qubit is referred to as“computing the logical-AND of the controls”, un-computing the ancillaqubit as “erasing the logical-AND”, and the combination of both piecesas a “temporary AND gate”.

Example Toffoli Constructions

FIG. 1 is a circuit representation 100 of a Toffoli construction. AToffoli gate is a universal reversible quantum logic gate. A Toffoligate acts on three qubits. If the first two qubits are in the state |1>,the Toffoli gate 102 flips the state of the third qubit and otherwiseleaves it unchanged. In FIG. 1 , the Toffoli gate 102 acts on threequbits represented by the three parallel horizontal lines 104, 106 and108. In FIG. 1 , qubits 104 and 106 represent control qubits, and qubit108 represents a target qubit. A Toffoli gate can be implemented usingthe construction 110. The construction 110 includes eight Cliffordgates—Controlled-NOT (CNOT) gates, e.g., CNOT gate 112, and Hadamardgates, e.g., Hadamard gate 114. The expanded circuit representation 110includes seven T gates, e.g., T gate 116 and 118.

Under the assumption that the construction 110 is not permitted toinvolve other qubits or to share work with other operations, thisconstruction was previously considered optimal. Otherwise, theconstruction may be optimized. For example, in cases where N adjacentToffoli gates share the same control qubits, the multiple Toffoli gatesmay be replaced by N−1 CNOT gates and one Toffoli gate. The T-count of Nadjacent Toffoli gates sharing the same controls is therefore 0N+O(1),where the marginal T-count is zero because each additional Toffoli canbe replaced with CNOTs framing a root Toffoli.

It may not be common for adjacent Toffoli gates to have the same controlqubits, however it may be common for a first Toffoli to later beun-computed by a second matching Toffoli, that is for the effect of thefirst Toffoli gate to be temporary. When this occurs, the three T gateson the control qubits of the construction shown above with reference toFIG. 1 can be omitted. In some cases, this may introduce phase errors.However the second Toffoli gate can un-compute those errors whileun-computing the state permutation.

Based on the Toffoli gate construction described with reference to FIG.1 , an n-bit quantum adder may contain 2n+O(1) Toffoli gates, which inturn implies a naive T-count of 14n+O(1). However, almost all of theToffoli gates in the first half of an adder are un-computed by Toffoligates in the second half. This allows the T gates on the controls of theToffoli gates to be omitted, reducing their T-count from 7 to 4 and theT-count of addition to 8n+O(1). Even if a Toffoli is not paired with asecond Toffoli that un-computes its effects, it is still possible toperform the Toffoli with T-count of 4 by using an ancilla qubit, ameasurement, and a conditional fixup operation.

Example Hardware

FIG. 2 depicts an exemplary system 200 for implementing temporaryToffoli gates and logical AND operations. The system 200 is an exampleof a system implemented as quantum or classical computer programs on oneor more quantum computing devices or classical computers in one or morelocations, in which the systems, components, and techniques describedbelow can be implemented.

The system 200 includes a quantum computing device 202 in datacommunication with one or more classical processors 204. The quantumcomputing device 202 includes components for performing quantumcomputation. For example, the quantum computing device 202 includes aquantum system 206, control devices 208, and T factories 210. Thequantum system 206 includes one or more multi-level quantum subsystems,e.g., a register of qubits. In some implementations the multi-levelquantum subsystems may be superconducting qubits, e.g., Gmon qubits. Thetype of multi-level quantum subsystems that the system 100 utilizes mayvary. For example, in some cases it may be convenient to include one ormore resonators attached to one or more superconducting qubits, e.g.,Gmon or Xmon qubits. In other cases ion traps, photonic devices orsuperconducting cavities (with which states may be prepared withoutrequiring qubits) may be used. Further examples of realizations ofmulti-level quantum subsystems include fluxmon qubits, silicon quantumdots or phosphorus impurity qubits.

Quantum circuits may be constructed and applied to the register ofqubits included in the quantum system 206 via multiple control linesthat are coupled to multiple control devices 208. Example controldevices 208 that operate on the register of qubits include quantum logicgates or circuits of quantum logic gates, e.g., Hadamard gates,controlled-NOT (CNOT) gates, controlled-phase gates, or T gates. In someimplementations T gates may be stored in one or more T factories 210included in the quantum computing device 202.

The control devices 208 may further include measurement devices, e.g.,readout resonators. Measurement results obtained via measurement devicesmay be provided to the classical processors 204 for processing andanalyzing.

Method for reusing |A

states

One innovative aspect of present disclosure describes a constructionthat improves the T-count of a single Toffoli gate by performing theToffoli indirectly instead of directly.

FIG. 3 is a flow diagram of an example process 300 for indirectlyperforming a Toffoli gate on two control qubits and a target qubit. Forconvenience, the process 300 will be described as being performed by aquantum computing device in communication with one or more classicalcomputing devices located in one or more locations. For example, thesystem 200 of FIG. 2 , appropriately programmed in accordance with thisspecification, can perform the process 300.

The system obtains an ancilla qubit in an A-state (step 302).

The system computes a logical-AND of the two control qubits and storesthe computed logical-AND in the state of the ancilla qubit by replacingthe A-state of the ancilla qubit with the logical-AND of the two controlqubits (step 304). To compute a logical-AND of the two control qubitsand store the computed logical-AND in the state of the ancilla qubit,the system may first apply a CNOT gate between the ancilla qubit in the|A> state and a first control qubit. The system may then apply theHermitian conjugate of a T gate to the ancilla qubit. The system maythen apply a CNOT gate between the ancilla qubit and a second controlqubit. The system may then apply a T gate to the ancilla qubit. Thesystem may then apply a CNOT gate between the ancilla qubit and thefirst control qubit. The system may then apply the Hermitian conjugateof a T gate to the ancilla qubit. The system may then apply a Hadamardgate to the ancilla qubit to store the logical AND of the two controlqubits in the state of the ancilla qubit. An example circuitrepresentation of computing a logical-AND of two control qubits andstoring the computed logical-AND in the state of an ancilla qubit isillustrated below with reference to FIG. 4 .

In some implementations the computation performed in step 304 mayintroduce phase errors. Such phase errors may be corrected by applying acontrolled-S quantum logic gate to the two control qubits. The effect ofan application of a controlled-S gate S=diag(1, e^(iπ/2)) to the twocontrol qubits is to apply a phase factor of i to the amplitudes ofcomputational basis states where both controls are on. Since the outputof the step 304 is a qubit whose state indicates whether or not bothcontrol qubits are on, the controlled-S gate on the two control qubitscan be replaced with an uncontrolled-S gate on the ancilla qubit storingthe logical-AND of the two control qubits.

The system applies a CNOT quantum logic gate between (i) the ancillaqubit storing the logical-AND of the two control qubits, and (ii) thetarget qubit, the ancilla qubit acting as a control qubit for the CNOTquantum logic gate (step 306).

The system provides the ancilla qubit storing the logical-AND of the twocontrol qubits as a resource for one or more additional operations (step308).

The system un-computes the logical-AND of the two control qubits andrecovers the A-state of the ancilla qubit by replacing the state of theancilla qubit storing the computed logical-AND of the two control qubitswith an A-state (step 310).

For example, the system may apply a Hadamard gate to the ancilla qubitstoring the logical-AND of the two control qubits. The system may thenapply a T gate to the ancilla qubit. The system may then apply a CNOTgate between the ancilla qubit and a first control qubit. The system maythen apply the Hermitian conjugate of a T gate to the ancilla qubit. Thesystem may then apply a CNOT gate between the ancilla qubit and a secondcontrol qubit. The system may then apply a T gate to the ancilla qubit.The system may then apply a CNOT gate between the ancilla qubit and thefirst control qubit to leave the ancilla qubit in an |A> state. Anexample circuit representation of un-computing a logical-AND of twocontrol qubits and recovering an A-state of an ancilla qubit isillustrated below with reference to FIG. 4 .

In cases where the system applies an S gate during the computation ofthe logical AND described above with reference to step 304, the systemmay apply the Hermitian adjoint of the S gate prior to applying thefirst Hadamard gate to the ancilla qubit storing the logical AND of thetwo control qubits.

The system provides the ancilla qubit in the recovered A-state as aresource for one or more additional operations (step 312). For example,the system may provide the ancilla qubit in the A-state to perform a Tgate.

In some implementations a first iteration of the process 300 may be usedto perform a first temporary Toffoli quantum logic gate on two firstcontrol qubits and a first target qubit. The system may then provide theancilla qubit in the recovered A-state as a resource for a seconditeration of the process 300 on two second control qubits and a secondtarget qubit.

FIG. 4 is an illustration 400 of an example quantum circuit forindirectly performing a Toffoli gate 402 on two control qubits and atarget qubit, as described above with reference to process 300 of FIG. 3. In illustration 400, a first control qubit of the two control qubitsis represented by horizontal line 406. A second control qubit of the twocontrol qubits is represented by horizontal line 404. Horizontal line408 represents the target qubit.

To perform the Toffoli gate 402 on the two control qubits 404, 406 andtarget qubit 408, an ancilla qubit, represented by horizontal line 409,in an A-state 410 is obtained. For example, the ancilla qubit 409 may beprepared in a 0-state. A Hadamard gate and T gate may be applied to theancilla qubit in the 0-state to obtain the A-state.

The logical AND of the two control qubits 404, 406 is computed 444 andstored in the state of the ancilla qubit 409. This includes applicationof: a CNOT gate 412 between the ancilla qubit 409 in the |A> state andthe first control qubit 406, the Hermitian conjugate of a T gate 414 tothe ancilla qubit 409, a CNOT gate 416 between the ancilla qubit 409 andthe second control qubit 404, a T gate 418 to the ancilla qubit 409, aCNOT gate 420 between the ancilla qubit 409 and the first control qubit406, the Hermitian conjugate of a T gate 422 to the ancilla qubit 409,and a Hadamard gate 424 to the ancilla qubit 409.

A CNOT quantum logic gate 426 is applied to (i) the ancilla qubit 409storing the logical-AND of the two control qubits 404, 406, and (ii) thetarget qubit 408, the ancilla qubit 409 acting as a control qubit forthe CNOT quantum logic gate 426.

The logical AND of the two control qubits 404, 406 is un-computed 446.This includes application of: a Hadamard gate 428 to the ancilla qubit409 storing the logical-AND of the two control qubits 404, 406, a T gate430 to the ancilla qubit 409, a CNOT gate 432 between the ancilla qubit409 and the first control qubit 406, the Hermitian conjugate of a T gate434 to the ancilla qubit 409, a CNOT gate 436 between the ancilla qubit409 and the second control qubit 404, a T gate 438 to the ancilla qubit409, and a CNOT gate 440 between the ancilla qubit 409 and the firstcontrol qubit 406. The ancilla qubit 409 may be returned to the 0-stateby application of the Hermitian conjugate of a T gate 442 and a Hadamardgate 448.

The indirect-Toffoli construction described with reference to FIGS. 3and 4 appears to have a T-count of 8. However, the last T gate 442 inthe circuit is unnecessary. In fact, in some cases it may be activelyharmful. By removing the T gate 442 and the following Hadamard 448, theT-count of the Toffoli gate is reduced from 8 to 7. In addition, theancilla qubit is left in an |A

state that can be consumed to perform a T gate elsewhere. This improvesthe net T-count of the Toffoli gate to 6.

This optimization construction may apply to multiple quantum circuits.For example, the optimization may be useful in circuits where an initialToffoli gate is later un-computed by a second Toffoli gate. Instead ofcomputing and un-computing the ancilla qubit for the first Toffoli gate,then re-computing and re-un-computing the ancilla qubit for the secondToffoli gate, the ancilla qubit may be maintained until the secondToffoli gate is un-computed. This halves the T-count of the pair from 12to 6, as illustrated in FIG. 5 .

FIG. 5 is an illustration 500 of an example quantum circuit forcomputing and un-computing multiple Toffoli gates 502 on two controlqubits 504, 506 and a target qubit 508 using an ancilla qubit 514. Asshown in illustration 500, maintaining the ancilla qubit until Toffoligate 512 is un-computed results in a net T-count of 6. Many circuitsinvolve computing and later un-computing a Toffoli, e.g., due toaddition operations. Previously, it was believed that each Toffoli gatehad a T-count of 4, the pair of Toffoli gates therefore having a T-countof 8. The process and construction described in FIGS. 3 and 4 reducesthe T-count of the pair from 8 to 6.

Constructing a Temporary AND

FIG. 6 is a flow diagram of an example process 600 for performing atemporary logical AND operation on two control qubits. For convenience,the process 600 will be described as being performed by a quantumcomputing device in communication with one or more classical computingdevices located in one or more locations. For example, the system 200 ofFIG. 2 , appropriately programmed in accordance with this specification,can perform the process 600.

The system obtains an ancilla qubit in an A-state (step 602).

The system computes a logical-AND of the two control qubits and storesthe computed logical-AND in the state of the ancilla qubit by replacingthe A-state of the ancilla qubit with the logical-AND of the two controlqubits (step 604). In some implementations, to correct any introducedphase errors, the system may further apply an uncontrolled-S gate to theancilla qubit. As described above with reference to FIG. 3 , computingthe logical-AND of the two control qubits includes applying three Tgates, optionally approximately in parallel.

The system maintains the ancilla qubit storing the logical-AND of thetwo controls until a first condition is satisfied (step 606). In someimplementations maintaining the ancilla qubit storing the logical-AND ofthe two control qubits may include providing the ancilla qubit as aresource for one or more additional operations, e.g., operations thatwould otherwise be conditioned on the two control qubits.

The system erases the ancilla qubit when the first condition issatisfied (step 608). For example, the system may erase the ancillaqubit when the one or more additional operations described above withreference to step 606 have been performed. In some implementationserasing the ancilla qubit may include transitioning the ancilla qubitinto a state that is independent of the state of the two control qubitsand does not cause the two control qubits to decohere.

To erase the ancilla qubit, the system may apply a measure-and-correctprocess that includes one or more Clifford operations (with no T-count)instead of un-computing the ancilla qubit using a mirror of the processused to compute the ancilla qubit. By starting with a process thatobviously performs the un-computation, e.g., as described above withreference to FIG. 3 , the process can be altered to generate themeasure-and-correct process. The un-computation process described inFIG. 3 includes a Toffoli gate, which clears the ancilla qubit since theancilla qubit was computed with a Toffoli gate and Toffoli gates aretheir own inverse. Since the cleared ancilla qubit is eventuallydiscarded, it is possible to apply a Hadamard gate and a measurement toit after the Toffoli gate but before discarding it. The Hadamard maythen be hopped over the Toffoli gate, transforming it into a CCZoperation. The CCZ may be rearranged so that the ancilla qubit is one acontrol qubit, which is possible because the control qubits and targetqubit of a CCZ are interchangeable. Finally, the deferred measurementprinciple may be invoked to hop the measurement over the CCZ, turningthe quantum control into a classical control. That is, to perform themeasure-and-correct process the system may apply a Hadamard quantumlogic gate to the ancilla qubit, measure the ancilla qubit to generate ameasurement result and analyze the generated measurement result. Inresponse to determining that the generated measurement result indicatesthat the two control qubits are both ON, the system may apply a CZ gateto the control qubits.

FIG. 7 is an illustration 700 of an example quantum circuit forperforming a temporary logical AND operation 701 on two control qubits702, 704. As shown in illustration 700, the computation of the logicalAND gate is drawn as an ancilla qubit wire 708 emerging vertically fromtwo controls then heading rightward.

Performing the temporary logical AND operation 701 includes obtaining anancilla qubit 706 in an A-state, applying a CNOT gate 710 between thefirst control qubit 702 and the ancilla qubit 706, applying a CNOT gate712 between the second control qubit 704 and the ancilla qubit 706,applying two CNOT gates 714 between the first control qubit 702, secondcontrol qubit 704 and ancilla qubit 706 (the order of which isirrelevant), applying a Hermitian conjugate of a T gate 716, 718 to thefirst control qubit 702 and to the second control qubit 704 and a T gate720 to the ancilla qubit 706 (the T gates 716, 718, 720 may be appliedapproximately in parallel), applying two CNOT gates 722 between thefirst control qubit 702, second control qubit 704 and ancilla qubit 706(the order of which is irrelevant), applying a Hadamard gate 724 to theancilla qubit 706 and, optionally, applying a S gate 726 to the ancillaqubit 706. The T count of the operation 701 is therefore 4 (includingthe T gate required to prepare the A-state of the ancilla qubit 706).

FIG. 8 is an illustration 800 of an example quantum circuit forun-computing a temporary logical AND operation 801 on two control qubits802, 804. As shown in illustration 800, the un-computation of thelogical AND gate 802 is drawn as an ancilla qubit wire 808 coming infrom the left then merging vertically into the two control qubits 802,804 that created it.

Un-computing the temporary logical AND operation 801 includes performinga measure-and-correct process. A Hadamard gate 810 is applied to theancilla qubit 806. The ancilla qubit 806 is measured 812. A CZ gate 814is applied to the control qubits 802, 804 if the generated measurementresult from measurement operation 812 indicates that the two controlqubits 802, 804 are both ON. The T count of the operation 801 is zero.

Applications

The above described processes and constructions may be used improve thecomplexity of several quantum circuits. For example, known adderconstructions such as the Cuccaro adder contain many Toffoli gates thatare later un-computed by another Toffoli. The presently describedprocesses and constructions do not fundamentally change the structure ofsuch known adders. Therefore, an improved adder construction based ontemporary Toffoli gates may be synthesized using the above describedtemporary AND gate construction, halving the T count of the adder.

FIG. 9 illustrates a per-bit building block 900 of an improved adderconstruction with a T count of 4. The building block 900 may be used aspart of a ripple-carry approach to performing addition, and can be usedto construct an n-bit adder by nesting n copies of the building block900 inside of each other. The compute carry bit box 902 representscomputation of the majority of the three input bits c_(k), i_(k), t_(k),that is whether or not the sum of the three bits will cause a carry intothe next bit k+1, and storing of the computed majority into the new wirethat will feed into the next bit's k+1 adder. The un-compute carry bitbox 904 represents the inverse operations of the majority computed inthe compute carry bit box 902, as well as operations for ensuring thatthe output bit has been toggled if the input bit is on.

FIG. 10 is an illustration a 5 bit adder with a T count of 16,constructed by tiling the per-bit building block 900 shown in FIG. 9 .Since the low bit corresponding to does not have a carry-in, the circuithas been optimized to omit that part. Also, since the high bit doesn'thave a carry-out, that has also been optimized to omit that part. Thebits corresponding to the input register are labelled i0, i1, i2, i3,i4. The target bits are labelled t0, t1, t2, t3, t4. After the circuithas been performed, the target bits have been modified such that thetarget register's new value is the sum of the input and the targetregister's old value, e.g., (t+i)1.

The building block 900 for the improved adder construction can bemodified such that the sum computed by the adder can be made availablefor use as soon as the carry signal hits the first control bit—insteadof needing to wait for the un-computation sweep to finish. This canhalve the T-count of the addition when it is going to be un-computed.Instead of using 4n+O(1) T gates to compute the addition, and then4n+O(1) more T gates to un-compute the addition, the intermediate stateof a single addition computation is utilized.

Furthermore, in some cases additions may be conditioned on a controlqubit, e.g., additions performed in Shor's algorithm. In some cases acontrolled-addition construction may have a T-count of 21n+O(1). Usingthe presently described temporary AND gate construction can improve thisto 8n+O(1).

The presently described temporary AND gate may also be applicable toother operations. For example, temporary AND operations can be usefulfor applying phase rotations to multiple qubits approximatelysimultaneously. Given a b-bit ancilla register G prepared in the state

$\left. {2^{{- b}/2}{\sum_{k = 0}^{2^{b} - 1}{e^{\frac{2i\pi k}{2^{b}}}{❘k}}}} \right\rangle$

(a ‘phase gradient state’), using the adder construction described aboveto add a register Q into G will cause phase kickback that applies theoperation

${\left. {{grad} = {\sum_{k = 0}^{2^{b} - 1}{e^{\frac{2i\pi k}{2^{b}}}{❘k}}}} \right\rangle\left\langle k \right.}❘$

to Q. The Grad operation is equivalent to applying the phase gateZ^(n″k) to the qubit position at k for each qubit within Q. Since somequantum Fourier transform circuits involve conditional uses of Grad,temporary-AND operations may be used to improve the T-count of thosecircuits.

In some cases it may be estimated that factoring a 2000-bit number maytake 27 hours and 2×10¹² distilled |A

states. This time estimate is based on each Toffoli having a T-depth of1, and the |A

state count estimate is based on Toffoli gates having a T-count of 7.The presently described processes and constructions reduce many existingestimates of the cost of quantum computation, improving thecomputational efficiency of quantum computations. For example, becauseShor's algorithm is dominated by the cost of additions, the presentlydescribed techniques multiply the T-count and T depth for factoring a2000-bit number by 4/14 and 1/3, respectively. This reduces theestimates to 9 hours and 6×10¹¹ distilled |A

states.

Other examples of operations implemented by quantum devices whichbenefit from cheaper temporary AND gates include but are not limited to:integer comparisons, integer multiplication, incrementing and counting,integer arithmetic in general, modular arithmetic, expanding a binaryregister into a unary register, operations with a target qubit indexedby a binary qubit register, phasing a register by a computable functionf (i.e. applying the operation), temporary permutations, or oracles inGrover's algorithm.

Implementations of the digital and/or quantum subject matter and thedigital functional operations and quantum operations described in thisspecification can be implemented in digital electronic circuitry,suitable quantum circuitry or, more generally, quantum computationalsystems, in tangibly-embodied digital and/or quantum computer softwareor firmware, in digital and/or quantum computer hardware, including thestructures disclosed in this specification and their structuralequivalents, or in combinations of one or more of them. The term“quantum computational systems” may include, but is not limited to,quantum computers, quantum information processing systems, quantumcryptography systems, or quantum simulators.

Implementations of the digital and/or quantum subject matter describedin this specification can be implemented as one or more digital and/orquantum computer programs, i.e., one or more modules of digital and/orquantum computer program instructions encoded on a tangiblenon-transitory storage medium for execution by, or to control theoperation of, data processing apparatus. The digital and/or quantumcomputer storage medium can be a machine-readable storage device, amachine-readable storage substrate, a random or serial access memorydevice, one or more qubits, or a combination of one or more of them.Alternatively or in addition, the program instructions can be encoded onan artificially-generated propagated signal that is capable of encodingdigital and/or quantum information, e.g., a machine-generatedelectrical, optical, or electromagnetic signal, that is generated toencode digital and/or quantum information for transmission to suitablereceiver apparatus for execution by a data processing apparatus.

The terms quantum information and quantum data refer to information ordata that is carried by, held or stored in quantum systems, where thesmallest non-trivial system is a qubit, i.e., a system that defines theunit of quantum information. It is understood that the term “qubit”encompasses all quantum systems that may be suitably approximated as atwo-level system in the corresponding context. Such quantum systems mayinclude multi-level systems, e.g., with two or more levels. By way ofexample, such systems can include atoms, electrons, photons, ions orsuperconducting qubits. In many implementations the computational basisstates are identified with the ground and first excited states, howeverit is understood that other setups where the computational states areidentified with higher level excited states are possible.

The term “data processing apparatus” refers to digital and/or quantumdata processing hardware and encompasses all kinds of apparatus,devices, and machines for processing digital and/or quantum data,including by way of example a programmable digital processor, aprogrammable quantum processor, a digital computer, a quantum computer,multiple digital and quantum processors or computers, and combinationsthereof. The apparatus can also be, or further include, special purposelogic circuitry, e.g., an FPGA (field programmable gate array), an ASIC(application-specific integrated circuit), or a quantum simulator, i.e.,a quantum data processing apparatus that is designed to simulate orproduce information about a specific quantum system. In particular, aquantum simulator is a special purpose quantum computer that does nothave the capability to perform universal quantum computation. Theapparatus can optionally include, in addition to hardware, code thatcreates an execution environment for digital and/or quantum computerprograms, e.g., code that constitutes processor firmware, a protocolstack, a database management system, an operating system, or acombination of one or more of them.

A digital computer program, which may also be referred to or describedas a program, software, a software application, a module, a softwaremodule, a script, or code, can be written in any form of programminglanguage, including compiled or interpreted languages, or declarative orprocedural languages, and it can be deployed in any form, including as astand-alone program or as a module, component, subroutine, or other unitsuitable for use in a digital computing environment. A quantum computerprogram, which may also be referred to or described as a program,software, a software application, a module, a software module, a script,or code, can be written in any form of programming language, includingcompiled or interpreted languages, or declarative or procedurallanguages, and translated into a suitable quantum programming language,or can be written in a quantum programming language, e.g., QCL orQuipper.

A digital and/or quantum computer program may, but need not, correspondto a file in a file system. A program can be stored in a portion of afile that holds other programs or data, e.g., one or more scripts storedin a markup language document, in a single file dedicated to the programin question, or in multiple coordinated files, e.g., files that storeone or more modules, sub-programs, or portions of code. A digital and/orquantum computer program can be deployed to be executed on one digitalor one quantum computer or on multiple digital and/or quantum computersthat are located at one site or distributed across multiple sites andinterconnected by a digital and/or quantum data communication network. Aquantum data communication network is understood to be a network thatmay transmit quantum data using quantum systems, e.g. qubits. Generally,a digital data communication network cannot transmit quantum data,however a quantum data communication network may transmit both quantumdata and digital data.

The processes and logic flows described in this specification can beperformed by one or more programmable digital and/or quantum computers,operating with one or more digital and/or quantum processors, asappropriate, executing one or more digital and/or quantum computerprograms to perform functions by operating on input digital and quantumdata and generating output. The processes and logic flows can also beperformed by, and apparatus can also be implemented as, special purposelogic circuitry, e.g., an FPGA or an ASIC, or a quantum simulator, or bya combination of special purpose logic circuitry or quantum simulatorsand one or more programmed digital and/or quantum computers.

For a system of one or more digital and/or quantum computers to be“configured to” perform particular operations or actions means that thesystem has installed on it software, firmware, hardware, or acombination of them that in operation cause the system to perform theoperations or actions. For one or more digital and/or quantum computerprograms to be configured to perform particular operations or actionsmeans that the one or more programs include instructions that, whenexecuted by digital and/or quantum data processing apparatus, cause theapparatus to perform the operations or actions. A quantum computer mayreceive instructions from a digital computer that, when executed by thequantum computing apparatus, cause the apparatus to perform theoperations or actions.

Digital and/or quantum computers suitable for the execution of a digitaland/or quantum computer program can be based on general or specialpurpose digital and/or quantum processors or both, or any other kind ofcentral digital and/or quantum processing unit. Generally, a centraldigital and/or quantum processing unit will receive instructions anddigital and/or quantum data from a read-only memory, a random accessmemory, or quantum systems suitable for transmitting quantum data, e.g.photons, or combinations thereof.

The essential elements of a digital and/or quantum computer are acentral processing unit for performing or executing instructions and oneor more memory devices for storing instructions and digital and/orquantum data. The central processing unit and the memory can besupplemented by, or incorporated in, special purpose logic circuitry orquantum simulators. Generally, a digital and/or quantum computer willalso include, or be operatively coupled to receive digital and/orquantum data from or transfer digital and/or quantum data to, or both,one or more mass storage devices for storing digital and/or quantumdata, e.g., magnetic, magneto-optical disks, optical disks, or quantumsystems suitable for storing quantum information. However, a digitaland/or quantum computer need not have such devices.

Digital and/or quantum computer-readable media suitable for storingdigital and/or quantum computer program instructions and digital and/orquantum data include all forms of non-volatile digital and/or quantummemory, media and memory devices, including by way of examplesemiconductor memory devices, e.g., EPROM, EEPROM, and flash memorydevices; magnetic disks, e.g., internal hard disks or removable disks;magneto-optical disks; CD-ROM and DVD-ROM disks; and quantum systems,e.g., trapped atoms or electrons. It is understood that quantum memoriesare devices that can store quantum data for a long time with highfidelity and efficiency, e.g., light-matter interfaces where light isused for transmission and matter for storing and preserving the quantumfeatures of quantum data such as superposition or quantum coherence.

Control of the various systems described in this specification, orportions of them, can be implemented in a digital and/or quantumcomputer program product that includes instructions that are stored onone or more non-transitory machine-readable storage media, and that areexecutable on one or more digital and/or quantum processing devices. Thesystems described in this specification, or portions of them, can eachbe implemented as an apparatus, method, or system that may include oneor more digital and/or quantum processing devices and memory to storeexecutable instructions to perform the operations described in thisspecification.

While this specification contains many specific implementation details,these should not be construed as limitations on the scope of what may beclaimed, but rather as descriptions of features that may be specific toparticular implementations. Certain features that are described in thisspecification in the context of separate implementations can also beimplemented in combination in a single implementation. Conversely,various features that are described in the context of a singleimplementation can also be implemented in multiple implementationsseparately or in any suitable sub-combination. Moreover, althoughfeatures may be described above as acting in certain combinations andeven initially claimed as such, one or more features from a claimedcombination can in some cases be excised from the combination, and theclaimed combination may be directed to a sub-combination or variation ofa sub-combination.

Similarly, while operations are depicted in the drawings in a particularorder, this should not be understood as requiring that such operationsbe performed in the particular order shown or in sequential order, orthat all illustrated operations be performed, to achieve desirableresults. In certain circumstances, multitasking and parallel processingmay be advantageous. Moreover, the separation of various system modulesand components in the implementations described above should not beunderstood as requiring such separation in all implementations, and itshould be understood that the described program components and systemscan generally be integrated together in a single software product orpackaged into multiple software products.

Particular implementations of the subject matter have been described.Other implementations are within the scope of the following claims. Forexample, the actions recited in the claims can be performed in adifferent order and still achieve desirable results. As one example, theprocesses depicted in the accompanying figures do not necessarilyrequire the particular order shown, or sequential order, to achievedesirable results. In some cases, multitasking and parallel processingmay be advantageous.

What is claimed is:
 1. A method for performing multiple Toffoli quantumlogic gates, the method comprising: for each Toffoli quantum logic gateof the multiple Toffoli quantum logic gates: replacing an A-state of anancilla qubit with a logical-AND of two control qubits; applying a CNOTquantum logic gate between (i) the ancilla qubit storing the logical-ANDof the two control qubits and (ii) a target qubit, the ancilla qubitacting as a control qubit for the CNOT quantum logic gate; recoveringthe A-state of the ancilla qubit by replacing the state of the ancillaqubit storing the logical-AND of the two control qubits with an A-state;and providing the ancilla qubit in the recovered A-state to perform asubsequent Toffoli quantum logic gate.
 2. The method of claim 1, furthercomprising providing the ancilla qubit in the A-state as a resource forone or more additional operations, wherein the one or more additionaloperations comprise performing a T gate.
 3. The method of claim 1,further comprising computing the logical-AND of the two control qubitsand uncomputing the logical-AND of the two control qubits, wherein thecomputing and uncomputing comprises performing six T gates.
 4. Themethod of claim 3, wherein computing the logical-AND of the two controlqubits comprises: applying a CNOT gate between the ancilla qubit in theA-state and a first one of the two control qubits; applying theHermitian conjugate of a T gate to the ancilla qubit; applying a CNOTgate between the ancilla qubit and a second one of the two controlqubits; applying a T gate to the ancilla qubit; applying a CNOT gatebetween the ancilla qubit and the first one of the two control qubits;applying the Hermitian conjugate of a T gate to the ancilla qubit; andapplying a Hadamard gate to the ancilla qubit to store the logical ANDof the two control qubits in the state of the ancilla qubit.
 5. Themethod of claim 4, further comprising applying a S gate to the ancillaqubit storing the logical-AND of the two control qubits.
 6. The methodof claim 1, wherein recovering the A-state of the ancilla qubit byreplacing the state of the ancilla qubit storing the logical-AND of thetwo control qubits with the A-state comprises: applying a Hadamard gateto the ancilla qubit storing the logical-AND of the two control qubits;applying a T gate to the ancilla qubit; applying a CNOT gate between theancilla qubit and a first one of the two control qubits; applying theHermitian conjugate of a T gate to the ancilla qubit; applying a CNOTgate between the ancilla qubit and a second one of the two controlqubits; applying a T gate to the ancilla qubit; and applying a CNOT gatebetween the ancilla qubit and the first one of the two control qubits toleave the ancilla qubit in the A-state.
 7. The method of claim 6,further comprising applying a S gate to the ancilla qubit.
 8. A quantumcomputing device comprising: a register of qubits comprising two controlqubits, a first target qubit, and an ancilla qubit prepared in aninitial state; a plurality of control lines coupled to the register ofqubits; a plurality of control circuits coupled to the plurality ofcontrol lines; and one or more computer-readable devices, includingtherein instructions that, when executed by one or more processors,cause the quantum computing device to perform operations for performingmultiple Toffoli quantum logic gates, wherein the operations include,for each Toffoli quantum logic gate of the multiple Toffoli quantumlogic gates: replacing an A-state of an ancilla qubit with a logical-ANDof two control qubits; applying a CNOT quantum logic gate between (i)the ancilla qubit storing the logical-AND of the two control qubits and(ii) a target qubit, the ancilla qubit acting as a control qubit for theCNOT quantum logic gate; recovering the A-state of the ancilla qubit byreplacing the state of the ancilla qubit storing the logical-AND of thetwo control qubits with an A-state; and providing the ancilla qubit inthe recovered A-state to perform a subsequent Toffoli quantum logicgate.
 9. A method for performing a temporary logical AND operation ontwo control qubits, the method comprising: replacing an A-state of anancilla qubit with a logical-AND of the two control qubits; andmaintaining the ancilla qubit storing the logical-AND of the twocontrols until a first condition is satisfied.
 10. The method of claim9, further comprising erasing the ancilla qubit when the first conditionis satisfied.
 11. The method of claim 9, wherein maintaining the ancillaqubit storing the logical-AND of the two controls until the firstcondition is satisfied comprises providing the ancilla qubit storing thelogical-AND of the two control qubits as a resource for one or moreadditional operations.
 12. The method of claim 11, wherein the one ormore additional operations comprise operations that would otherwise beconditioned on the two control qubits.
 13. The method of claim 11,wherein erasing the ancilla qubit when the first condition is satisfiedcomprises erasing the ancilla qubit when the one or more additionaloperations have been performed.
 14. The method of claim 9, whereinerasing the ancilla qubit comprises transitioning the ancilla into astate that is independent of the state of the two control qubits anddoes not cause the two control qubits to decohere.
 15. The method ofclaim 10, wherein erasing the ancilla qubit comprises applying ameasure-and-correct process.
 16. The method of claim 15, wherein themeasure-and-correct process comprises: applying a Hadamard quantum logicgate to the ancilla qubit; measuring the ancilla qubit to generate ameasurement result; and in response to determining that the generatedmeasurement result indicates that the two control qubits are both ON,applying a CZ gate.
 17. The method of claim 15, wherein themeasure-and-correct process comprises Clifford operations.
 18. Themethod of claim 9, further comprising correcting phase errors byapplying an uncontrolled S gate to the ancilla qubit.
 19. The method ofclaim 9, further comprising computing the logical-AND of the two controlqubits, wherein computing the logical-AND of the two control qubitscomprises applying three T gates, optionally approximately in parallel.20. A quantum computing device comprising: a register of qubitscomprising two control qubits, a target qubit, and an ancilla qubitprepared in an initial state; a plurality of control lines coupled tothe register of qubits; a plurality of control circuits coupled to theplurality of control lines; and one or more computer-readable devices,including therein instructions that, when executed by one or moreprocessors, cause the quantum computing device to perform operations forperforming a temporary logical AND operation on two control qubits,wherein the operations include: replacing an A-state of the ancillaqubit with a logical-AND of the two control qubits; and maintaining theancilla qubit storing the logical-AND of the two controls until a firstcondition is satisfied.